n - 1 (a - θ)2
max ∏npv = —— +--+
θ n+ 1 4b
(1 - c )θ (a2bθ)
1 + δ
(16)
The optimal fee is equal to27
θ* = a (1 -
2v
1+δ____)
2 - (1 + δ))
n+1
(17)
We can now show how the monitoring cost and the default risk affect
the choice of system of rent extraction.
Proposition 1 The higher (lower) the monitoring cost, c, and default
risk, δ, the lower (higher) is the fee θ, i.e., the more is up-front (ex-post)
collection used.
Proof. The derivatives
2(1-C )
■ = - a .....n-1_______< 0
dδ 2(2⅛ - (1+ δ))2
n + 1
and
∂θ∙ = _ n+1(1 + δ) < 0
∂c at(2(1 - c) + (1 + δ)n+1 )2
prove the proposition. ■
When shaping the rent-extraction system, the CA weighs the costs
and benefits of up-front collection relative to ex-post collection. The
cost of using up-front collection is that the full value of the office cannot
be collected due to the auction mechanism. The cost of using ex-post
payments is that the agent might steal and that it is costly to borrow.
A high monitoring cost, c, generates much theft such that the ex-post
revenues and the amount that can be borrowed are low. Similarly, only a
small amount can be borrowed when the default risk, δ, is high. In both
1 c
27For θ* > 0, it is necessary that 1+δ > n-t-.
n+1
19